Unschooling Math Suggestions
mommaofaton
Hello, I'm searching for suggestions, tips, or ideas for unschooling
math. Any suggestions would be appreciated. Thank you.
Kay Beth
Country Momma
math. Any suggestions would be appreciated. Thank you.
Kay Beth
Country Momma
diana jenner
On 6/14/07, mommaofaton <mommaofaton@...> wrote:
biting and non-judgmental discussions. My kids could very quickly figure out
% for discounts and % for tax. Hayden, at 8.7, plans his budget for every
month (he prefers his money in lump sum, though we've had conversations
about how much that is per day & per week & bi-weekly, he prefers monthly).
Sticks to it or not, the money is truly his own. My job is to sometimes
remind him (his request) for things farther down the calendar that he may
want some cash for.
The best proof to my mother-in-law of the effectiveness of this idea, was
when we were shopping in Los Angeles and Hayden, then 6, asked her the sales
tax rate (raised eyebrows, he knows about tax??) and was then horrified at
the 9.5% response, "I have to give nearly 10 cents of every dollar to the
state??" :::vbg::: I have a feeling his growing affinity for Oregon may have
a little something to do with their lack of sales tax!!
Most boardgames have math and counting -- if you need to hold that in your
head while you play -- but please don't play *just because* there's math
involved, nor manipulate them into playing because you think they need math.
Play! Math sneaks in because it's everywhere!
--
~diana :)
xoxoxoxo
hannahbearski.blogspot.com
[Non-text portions of this message have been removed]
>Money! Their own no-strings-attached cash. It requires a lot of tongue
> Hello, I'm searching for suggestions, tips, or ideas for unschooling
> math. Any suggestions would be appreciated. Thank you.1
>
>
>
>
biting and non-judgmental discussions. My kids could very quickly figure out
% for discounts and % for tax. Hayden, at 8.7, plans his budget for every
month (he prefers his money in lump sum, though we've had conversations
about how much that is per day & per week & bi-weekly, he prefers monthly).
Sticks to it or not, the money is truly his own. My job is to sometimes
remind him (his request) for things farther down the calendar that he may
want some cash for.
The best proof to my mother-in-law of the effectiveness of this idea, was
when we were shopping in Los Angeles and Hayden, then 6, asked her the sales
tax rate (raised eyebrows, he knows about tax??) and was then horrified at
the 9.5% response, "I have to give nearly 10 cents of every dollar to the
state??" :::vbg::: I have a feeling his growing affinity for Oregon may have
a little something to do with their lack of sales tax!!
Most boardgames have math and counting -- if you need to hold that in your
head while you play -- but please don't play *just because* there's math
involved, nor manipulate them into playing because you think they need math.
Play! Math sneaks in because it's everywhere!
--
~diana :)
xoxoxoxo
hannahbearski.blogspot.com
[Non-text portions of this message have been removed]
[email protected]
I second Diana's suggestion of money and board games ! It's real life and it
has meaning for them.
We also have 2 books that we keep out...that are always in use by my kids:
-Barron's Math Wizardry for Kids Margaret Kenda and Phyllis Williams
Solve puzzles,play games, have fun! Surprise yourself with your own wizardry.
- 1000 Play Thinks by Ivan Moscovich
Puzzles, paradoxes, Illusions & Games.
marcia in MA
JUSTLIVELIFE
************************************** See what's free at http://www.aol.com
[Non-text portions of this message have been removed]
has meaning for them.
We also have 2 books that we keep out...that are always in use by my kids:
-Barron's Math Wizardry for Kids Margaret Kenda and Phyllis Williams
Solve puzzles,play games, have fun! Surprise yourself with your own wizardry.
- 1000 Play Thinks by Ivan Moscovich
Puzzles, paradoxes, Illusions & Games.
marcia in MA
JUSTLIVELIFE
************************************** See what's free at http://www.aol.com
[Non-text portions of this message have been removed]
Backstrom kelli
Sadie (my almost 7 year old ) loves math which is always surprising to me because me and dh hate it:)
She also just loves to spend time with us so we play tic tac toe with different problems in the squares, some harder than others and give her an allowance and let her spend it. She plays with one of those leap frog math things my in laws gave her and she has a couple little workbooks that she loves.
A lot of un unschooly suggestions but this is my daughter and these are the ways she choses to learn about the way math works in her world.
Also talking to her a lot! She is always asking me questions about math.
Kelli
diana jenner <hahamommy@...> wrote: On 6/14/07, mommaofaton <mommaofaton@...> wrote:
biting and non-judgmental discussions. My kids could very quickly figure out
% for discounts and % for tax. Hayden, at 8.7, plans his budget for every
month (he prefers his money in lump sum, though we've had conversations
about how much that is per day & per week & bi-weekly, he prefers monthly).
Sticks to it or not, the money is truly his own. My job is to sometimes
remind him (his request) for things farther down the calendar that he may
want some cash for.
The best proof to my mother-in-law of the effectiveness of this idea, was
when we were shopping in Los Angeles and Hayden, then 6, asked her the sales
tax rate (raised eyebrows, he knows about tax??) and was then horrified at
the 9.5% response, "I have to give nearly 10 cents of every dollar to the
state??" :::vbg::: I have a feeling his growing affinity for Oregon may have
a little something to do with their lack of sales tax!!
Most boardgames have math and counting -- if you need to hold that in your
head while you play -- but please don't play *just because* there's math
involved, nor manipulate them into playing because you think they need math.
Play! Math sneaks in because it's everywhere!
--
~diana :)
xoxoxoxo
hannahbearski.blogspot.com
[Non-text portions of this message have been removed]
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Get the free Yahoo! toolbar and rest assured with the added security of spyware protection.
[Non-text portions of this message have been removed]
She also just loves to spend time with us so we play tic tac toe with different problems in the squares, some harder than others and give her an allowance and let her spend it. She plays with one of those leap frog math things my in laws gave her and she has a couple little workbooks that she loves.
A lot of un unschooly suggestions but this is my daughter and these are the ways she choses to learn about the way math works in her world.
Also talking to her a lot! She is always asking me questions about math.
Kelli
diana jenner <hahamommy@...> wrote: On 6/14/07, mommaofaton <mommaofaton@...> wrote:
>Money! Their own no-strings-attached cash. It requires a lot of tongue
> Hello, I'm searching for suggestions, tips, or ideas for unschooling
> math. Any suggestions would be appreciated. Thank you.1
>
>
>
>
biting and non-judgmental discussions. My kids could very quickly figure out
% for discounts and % for tax. Hayden, at 8.7, plans his budget for every
month (he prefers his money in lump sum, though we've had conversations
about how much that is per day & per week & bi-weekly, he prefers monthly).
Sticks to it or not, the money is truly his own. My job is to sometimes
remind him (his request) for things farther down the calendar that he may
want some cash for.
The best proof to my mother-in-law of the effectiveness of this idea, was
when we were shopping in Los Angeles and Hayden, then 6, asked her the sales
tax rate (raised eyebrows, he knows about tax??) and was then horrified at
the 9.5% response, "I have to give nearly 10 cents of every dollar to the
state??" :::vbg::: I have a feeling his growing affinity for Oregon may have
a little something to do with their lack of sales tax!!
Most boardgames have math and counting -- if you need to hold that in your
head while you play -- but please don't play *just because* there's math
involved, nor manipulate them into playing because you think they need math.
Play! Math sneaks in because it's everywhere!
--
~diana :)
xoxoxoxo
hannahbearski.blogspot.com
[Non-text portions of this message have been removed]
---------------------------------
Get the free Yahoo! toolbar and rest assured with the added security of spyware protection.
[Non-text portions of this message have been removed]
Fetteroll
On Jun 14, 2007, at 11:20 AM, mommaofaton wrote:
but as a tool that is used while living life.
You don't need to unschool math any more than you need to unschool
the English language.
I think the biggest problem, though, is that schools spend so much
time doing math that it seems like the little bit that kids do in
real life can't possibly be enough.
The reason schools spend so much time is because they go about it
backwards and it's hard. They begin with the abstract and hope the
kids will extract the meaning from it. Abstract is *hard* when you
don't have a good feeling for what it is. Unfortunately, since
schools need to show that kids are learning, they have to teach in a
way that's testable. You can't test concepts that are being absorbed
in odd ways and refined as they're being used. Imagine trying to
devise a standardized test for 2 year olds to measure their progress
in English. My daughter could say pachycephalosaurus before she could
say her own name! There isn't a test in the world that would have
given a good picture of her unique progress through mastering
English ;-)
Learning math is as easy as learning a native language. Kids absorb
and refine the concepts because they use them.
My daughter uses math for shopping, tips, video games, art software,
"How long until ...", board and card games, Undoubtedly much more but
those are the ones she encountered most often.
Joyce
[Non-text portions of this message have been removed]
> I'm searching for suggestions, tips, or ideas for unschoolingIt helps loads to not see math as something alien that needs learned
> math.
but as a tool that is used while living life.
You don't need to unschool math any more than you need to unschool
the English language.
I think the biggest problem, though, is that schools spend so much
time doing math that it seems like the little bit that kids do in
real life can't possibly be enough.
The reason schools spend so much time is because they go about it
backwards and it's hard. They begin with the abstract and hope the
kids will extract the meaning from it. Abstract is *hard* when you
don't have a good feeling for what it is. Unfortunately, since
schools need to show that kids are learning, they have to teach in a
way that's testable. You can't test concepts that are being absorbed
in odd ways and refined as they're being used. Imagine trying to
devise a standardized test for 2 year olds to measure their progress
in English. My daughter could say pachycephalosaurus before she could
say her own name! There isn't a test in the world that would have
given a good picture of her unique progress through mastering
English ;-)
Learning math is as easy as learning a native language. Kids absorb
and refine the concepts because they use them.
My daughter uses math for shopping, tips, video games, art software,
"How long until ...", board and card games, Undoubtedly much more but
those are the ones she encountered most often.
Joyce
[Non-text portions of this message have been removed]
Ren Allen
~~You don't need to unschool math any more than you need to unschool
the English language.~~
Exactly.
Seeing the world in subjects is an impediment to joyful unschooling.
When math and history and science and everything else schools try to
divide, are simply part and parcel of pursuing passions (yeah, I'm
having fun with that one today), then subjects become irrelevant. The
interest and fascination humans have with different topics and
information will involve math, as we follow those interests.
We don't need to worry about getting math into our children. We need
to focus on living a rich and full life.
It IS good to be aware if we ourselves have hangups and damage about
math (or anything else) so we don't automatically shun things that are
related to it. I found that as I healed old school wounds, I saw math
in a totally different light. Math is everywhere. It's in art, music
and daily life. It can't be avoided.
Ren
learninginfreedom.com
the English language.~~
Exactly.
Seeing the world in subjects is an impediment to joyful unschooling.
When math and history and science and everything else schools try to
divide, are simply part and parcel of pursuing passions (yeah, I'm
having fun with that one today), then subjects become irrelevant. The
interest and fascination humans have with different topics and
information will involve math, as we follow those interests.
We don't need to worry about getting math into our children. We need
to focus on living a rich and full life.
It IS good to be aware if we ourselves have hangups and damage about
math (or anything else) so we don't automatically shun things that are
related to it. I found that as I healed old school wounds, I saw math
in a totally different light. Math is everywhere. It's in art, music
and daily life. It can't be avoided.
Ren
learninginfreedom.com
Paul D. Fernhout
This is something I prepared for another list, but it might be useful
here. These resources are more for teens than younger ages (with a focus
on algebra, trigonometry, and higher math). They are mostly software
tools, but I also mention some videos, web sites, and competitions.
Mastering any of these software tools would provide quite a mathematical
education in an homeschooling sort of way -- if the child wanted to do that.
Here is a link to a discussion about "Open Source Math Software For
Education":
http://ask.slashdot.org/article.pl?sid=04/12/13/2355258
It discusses computer software tools which does mathematical things.
In general, these sorts of tools can be wonderful playthings if you
and/or your child have the time and inclination to wrestle with them.
Another summary link:
"Software for Mathematics Education"
http://math.tkk.fi/~arasila/mathedu.html
Here is another link to a discussion of three big free software packages:
http://math-blog.com/2007/06/02/3-awesome-free-math-programs/
The big three mentioned there are:
Scilab
http://www.scilab.org/
"Scilab is a scientific software package for numerical computations
providing a powerful open computing environment for engineering and
scientific applications."
See also the related Scicos:
http://www.scicos.org/
"Scicos is a graphical dynamical system modeler and simulator toolbox
included in the Scilab ® engineering and scientific computation
software. With Scicos you can create block diagrams to model and
simulate the dynamics of hybrid dynamical systems and compile your
models into executable code. Scicos is used for signal processing,
systems control, queuing systems, and to study physical and biological
systems. New extensions allow generation of component based modeling of
electrical and hydraulic circuits using the Modelica language."
Maxima
http://maxima.sourceforge.net/
"Maxima is a system for the manipulation of symbolic and numerical
expressions, including differentiation, integration, Taylor series,
Laplace transforms, ordinary differential equations, systems of linear
equations, polynomials, and sets, lists, vectors, matrices, and tensors.
Maxima yields high precision numeric results by using exact fractions,
arbitrary precision integers, and arbitrarily precision floating point
numbers. Maxima can plot functions and data in two and three dimensions."
R
http://www.r-project.org/
"R is a free software environment for statistical computing and
graphics. It compiles and runs on a wide variety of UNIX platforms,
Windows and MacOS."
Here is another free package discussed at the first link:
http://www.gnu.org/software/octave/
"GNU Octave is a high-level language, primarily intended for numerical
computations. It provides a convenient command line interface for
solving linear and nonlinear problems numerically, and for performing
other numerical experiments using a language that is mostly compatible
with Matlab."
Here is another (simpler) free package for statistics and plotting:
http://www.gnuplot.info/
"Gnuplot is a portable command-line driven interactive data and
function plotting utility for UNIX, IBM OS/2, MS Windows, DOS,
Macintosh, VMS, Atari and many other platforms. "
However, to get a lot out of any of them, you have to put some time into
them and probably consult additional reference materials in their
tutorial manuals, on the web, and/or in related textbooks. Also, these
tools are all are a matter of convenience -- a general purpose
programming tool with some good libraries (e.g. Python + SciPy)
http://www.scipy.org/
http://www.scipy.org/Topical_Software
can let you do pretty much anything these specialized packages do and in
a much more expandable way -- it just takes *much* more time and
understanding to work with the general tools (though you learn a lot
doing that). I'd suggest perhaps doing both -- play with the tools *and*
try to learn how to program with various packages. :-)
As you can see from especially the third link above, there are several
other tools. I'm not sure how well all of these run under other
operating systems (most probably do run under Windows, but my wife and I
don't use that ourselves anymore except under emulation on the Mac or
Linux). You can probably find most of theses tools included (or easily
loadable) in Ubuntu (a version of GNU/Linux), which is available for
free here (you can download and burn a bootable CD to run these when you
want by rebooting, even if you run Windows):
http://www.ubuntu.com/
and they also mail out free CDs (but it may take some time -- see the
links on that page). There is also an "Edubuntu" educational variant,
but it seems more targeted at schools and I haven't tried it.
Instructions for loading some of these packages into Ubuntu:
http://ubuntuforums.org/showthread.php?s=c8e173f81d97693ff7bb02ad5cefba0a&t=402802&highlight=Mathematical+software
Any one of these packages is a project to get to know and understand. It
might takes quite a while to get a feel for them. But I think mastering
any of these packages will have a lot of practical value -- more so than
doing some paper exercises.
While our child is only three (and so I haven't tried to use these
packages to teach him math; I'm still focusing on John Holt's ideas of
seeing how small numbers like nine can be broken down and assembled many
ways), I have been involved tangentially with some people doing
mathematical education using computer software for more advanced math,
primarily using the programming language Python.
http://www.python.org/
There is a newer way of thinking about math and physics which suggests
thinking about that as just a subpart of computer programming, rather
than a more classical school subject approach which often puts computer
programming as an add-on under math & physics. :-) Essentially, by
developing programs and simulations of interest to a child, a child
learns more and more about basic principles of mathematics they need to
solve problems of interest to them, as well as to develop a certain type
of mathematical thinking skills related to precision and causality and
symbol manipulation and so on. Learning to program then becomes the
stepping stone to learning about math and physics.
For example, a child can learn a lot about trigonometry by just playing
around with a 2D or 3D turtle:
"turtle -- Turtle graphics for Tk using Python"
http://docs.python.org/lib/module-turtle.html
However they typically need to learn a little programming first (like
how to write a loop). Consider, for example, this college assignment:
"Mandelbrot, Turtle Graphics and L-Systems in Python"
http://www.cs.unc.edu/Courses/comp144-s02/assignments/2/index.html
I would think those sorts of experiments could keep a high schooler busy
for quite a while, while also learning about angles and distances and
mathematical reasoning (while they debug their graphical programs); but
I wouldn't try those as a first step for someone who did not already
know programming. Here is a simple code snippet for someone wanting to
learn some simple programming and turtle graphics with Python:
"Turtle Graphics (Python)"
http://www.daniweb.com/code/snippet304.html
Modify that to draw a few geometric shapes (cubes, octagons,
parallelograms, etc.) and it's a start down the path of 2D geometry. :-)
Here is a bigger set of links (mostly programs) related to
Python+Geometry for homeschoolers
http://www.4dsolutions.net/ocn/pygeom.html
In general the author of that site has a lot to say about the
intersection of computing and mathematics.
To get more advanced in 3D geometry, here is a free package by the late
Arthur Siegel to do interactive 3D geometry in Python:
http://pygeo.sourceforge.net/
"PyGeo is most fundamentally a framework for the creation of dynamic
geometric constructions - i.e. constructions which embody defined and
persistent geometric relationships responsive to real time on-screen
interactivity. PyGeo is, further, an implementation of this underlying
abstract framework - exposing a range of geometric objects as the
building blocks for virtual, dynamic geometric constructions. The focus
is away from Euclidian geometry and metrics, and toward later geometric
and mathematical developments - particularly those connected with
projective geometry of real space, and the geometry of complex numbers
on the plane and on the unit (Riemann) sphere."
Nothing about school-style trigonometry "proofs" there though -- except
the ones you build for yourself in passing to debug things. Here's lots
of links related to logic software and logic education if you want to
focus on proofs (includes some links to theorem provers):
http://www.cs.otago.ac.nz/staffpriv/hans/logiccourseware.html
But I would think this a dustier and drier route for most kids to get an
intuition for geometrical things than indirectly via programming.
Some fraction of the population has trouble visualizing or
rotating things in 2D or 3D,
http://en.wikipedia.org/wiki/Spatial_Visualization_Ability
so some kids find geometry much easier than others. I think 3D spatial
visualization tools may help narrow that gap, especially tools which can
make the 3d object rotate (at least slightly). Related exercises:
"Spatial Encounters: Exercises in Spatial Awareness."
http://www.womentechstore.com/edpw031-spatial-encounters-introduction.html
http://eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED236183&ERICExtSearch_SearchType_0=eric_accno&accno=ED236183
Likewise some kids almost need to visualize things to feel they
understand them as opposed to accept purely symbolic representations;
there are various approaches to mathematics that are more highly visual
than others. Example:
http://www.ms.uky.edu/~lee/visual05/visual05.html
If there is one issue that may make or break in some advanced math
studies, it is an ability to visualize. If the kid in question isn't
good at that, it may help more to focus early on visualization skills
(including blueprint reading) than to move ahead to more advanced topics.
Here is another programming language system Squeak (Smalltalk) which has
various educational tools in it like turtles:
http://www.squeak.org/Features/
It has a steeper learning curve than Python though.
In my own life I've found the only major use for trigonometry was
writing 3D graphics software. :-) Here is a example of such software
(available for free) which my wife and I wrote:
http://www.kurtz-fernhout.com/PlantStudio/index.htm
"PlantStudio Botanical Illustration Software is a tool for creating 3D
plant models and 2D illustrations. PlantStudio simulates herbaceous
(non-woody) plants like wildflowers and cut flowers, vegetables, weeds,
grasses, and herbs using a parameter-driven simulation of plant growth
and structure. You can "grow" plants over their life cycles, producing
lifelike images at any age. You can design, animate and breed a wide
variety of plants. By using the "evolutionary arts" of variation and
selection in the plant breeder, you can quickly and easily create whole
families of unique plants for your 3D scenes."
There's a lot of stuff in there about angles and rotations and so on --
can't help but learn a little about trigonometry (and botany) even by
just playing with it.
While I know schools tend to focus on a chain of algebra, trigonometry,
calculus, (especially for kids interested in engineering) I'd suggest
learning about probability and statistics (done as part of developing a
computer simulation) would generally be a more useful mathematical skill
in the real world for most people, since we all need to make decisions
every day related to risks and also trying to read between the lines of
newspaper articles and advertisements throwing statistics around in an
attempt to inform or persuade us.
For example, here is a fun link about how to lie with statistics:
http://faculty.washington.edu/chudler/stat3.html
"Ok...this is what you have been waiting for. How can you lie with
statistics? Actually, the purpose of this page is NOT to teach you how
to lie and cheat with statistics. Rather, I hope you will learn how it
is possible to be misled and how to spot "statistical abuse." You can
find poor use of statistics everywhere...magazines, newspapers, polls,
TV, even research papers. I do not want to hear of any of you readers
using these poor methods."
And here is a book on that topic:
_How to Lie With Statistics_
http://www.amazon.com/How-Lie-Statistics-Darrell-Huff/dp/0393310728
There are also a variety of free instructional videos available over the
web as part of the Annenberg CPB project if you have a fast internet
connection (and otherwise likely available from a local library or
through interlibrary loan):
http://www.learner.org/resources/browse.html
Exmaples:
"Against All Odds: Inside Statistics; This video instructional series
for college and high school classrooms and adult learners leads to a
greater understanding of statistics by exploring authentic examples —
from environmental studies to weight-loss programs."
"Algebra: In Simplest Terms; This video instructional series for college
and high school classrooms and adult learners guides students
step-by-step through algebra concepts, while highlighting common trouble
spots."
"Learning Math: Geometry; Learn the basics of geometry in this video-
and Web-based course for K-8 teachers."
I haven't watched any of those, but many years ago I did go through the
entire Mechanical Universe physics series for myself and liked it:
"The Mechanical Universe…and Beyond; This video instructional series
for college and high school classrooms and adult learners demystifies
physics and illustrates abstract concepts."
(Physics ends up being mostly math at more advanced stages.)
I also enjoyed their World of Chemistry, but I didn't see that is
online. There is also an accompanying textbook to the Mechanical
Universe; if anyone wanted something like a curriculum, that might
be a way to go which integrates mathematics into a physical setting that
helps motivate its study -- but you can also just watch the videos
for fun.
Here is another general resource for math education:
http://mathforum.org/
"The Math Forum is the leading online resource for improving math
learning, teaching, and communication since 1992.
_We are teachers, mathematicians, researchers, students, and parents
using the power of the Web to learn math and improve math education.
_We offer a wealth of problems and puzzles; online mentoring; research;
team problem solving; collaborations; and professional development.
Students have fun and learn a lot. Educators share ideas and acquire new
skills."
If your kid is a general theoretical math whiz
there is quite a bit of money to be won solving these incredible
difficult puzzles:
http://www.claymath.org/millennium/
"In order to celebrate mathematics in the new millennium, The Clay
Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven
Prize Problems. The Scientific Advisory Board of CMI selected these
problems, focusing on important classic questions that have resisted
solution over the years. The Board of Directors of CMI designated a $7
million prize fund for the solution to these problems, with $1 million
allocated to each."
It may even be fun just to look at the puzzles and think about what you
would need to know to even think about solving them. :-)
Here is a step down :-) -- the local William Lowell Putnam competition:
http://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition
http://math.scu.edu/putnam/
"Each year the annual Putnam Competition is held on the first Saturday
in December. ... The competition is open only to regularly enrolled
undergraduates, in colleges and universities of the United States and
Canada, who have not yet received a college degree. No individual may
participate in the competition more than four times. An eligible entrant
who is also a high school student must be informed of this four time limit."
Not much money even if you win the competition, but
perhaps a bit of glory. And something motivating for some -- to do well
in that contest. (Well, it helps that math departments sometimes pay for
a free lunch for the contestants. :-) Although ultimately the best
motivation comes from within for the love of the subject for itself.
Perhaps a competition might even be off-putting for some.
Anyway, even if you don't want to do the competition, the support
materials and sample questions may be of interest.
I'm a John Holt fan. Here is one thing he says about math:
http://www.naturalchild.org/guest/marlene_bumgarner.html
"What your philosophy about math?
...
The best way to meet numbers is in real life, as everything else. It’s
embedded in the context of reality, and what schooling does is to try to
take everything out of the context of reality. So everything appears
like some little thing floating around in space, and it’s a terrible
mistake. You know, there are numbers in building; there are numbers in
construction; there are numbers in business; there are numbers in
photography; there are numbers in music; there are fractions in cooking.
So wherever numbers are in real life, then let’s go and meet them and
work with them.
What subject matter do you see as essential?
None."
There is too much mathophobia in the world as it is -- in large part as
a consequence of how math is taught in schools:
http://www.google.com/search?hl=en&q=mathophobia
Yet higher math can be a very fun endeavor if done for its own sake.
As much as I like mathematical ideas, there are a lot of very successful
people in life who cannot do math at all. If your kid has other
interests, then advanced math is about as important as advanced zoology
-- probably less so. :-)
Anyway, plenty of open-ended free resources to play with and learn
something mathematical for kids or adults.
--Paul Fernhout
(Just joined the group.)
Fetteroll wrote:
here. These resources are more for teens than younger ages (with a focus
on algebra, trigonometry, and higher math). They are mostly software
tools, but I also mention some videos, web sites, and competitions.
Mastering any of these software tools would provide quite a mathematical
education in an homeschooling sort of way -- if the child wanted to do that.
Here is a link to a discussion about "Open Source Math Software For
Education":
http://ask.slashdot.org/article.pl?sid=04/12/13/2355258
It discusses computer software tools which does mathematical things.
In general, these sorts of tools can be wonderful playthings if you
and/or your child have the time and inclination to wrestle with them.
Another summary link:
"Software for Mathematics Education"
http://math.tkk.fi/~arasila/mathedu.html
Here is another link to a discussion of three big free software packages:
http://math-blog.com/2007/06/02/3-awesome-free-math-programs/
The big three mentioned there are:
Scilab
http://www.scilab.org/
"Scilab is a scientific software package for numerical computations
providing a powerful open computing environment for engineering and
scientific applications."
See also the related Scicos:
http://www.scicos.org/
"Scicos is a graphical dynamical system modeler and simulator toolbox
included in the Scilab ® engineering and scientific computation
software. With Scicos you can create block diagrams to model and
simulate the dynamics of hybrid dynamical systems and compile your
models into executable code. Scicos is used for signal processing,
systems control, queuing systems, and to study physical and biological
systems. New extensions allow generation of component based modeling of
electrical and hydraulic circuits using the Modelica language."
Maxima
http://maxima.sourceforge.net/
"Maxima is a system for the manipulation of symbolic and numerical
expressions, including differentiation, integration, Taylor series,
Laplace transforms, ordinary differential equations, systems of linear
equations, polynomials, and sets, lists, vectors, matrices, and tensors.
Maxima yields high precision numeric results by using exact fractions,
arbitrary precision integers, and arbitrarily precision floating point
numbers. Maxima can plot functions and data in two and three dimensions."
R
http://www.r-project.org/
"R is a free software environment for statistical computing and
graphics. It compiles and runs on a wide variety of UNIX platforms,
Windows and MacOS."
Here is another free package discussed at the first link:
http://www.gnu.org/software/octave/
"GNU Octave is a high-level language, primarily intended for numerical
computations. It provides a convenient command line interface for
solving linear and nonlinear problems numerically, and for performing
other numerical experiments using a language that is mostly compatible
with Matlab."
Here is another (simpler) free package for statistics and plotting:
http://www.gnuplot.info/
"Gnuplot is a portable command-line driven interactive data and
function plotting utility for UNIX, IBM OS/2, MS Windows, DOS,
Macintosh, VMS, Atari and many other platforms. "
However, to get a lot out of any of them, you have to put some time into
them and probably consult additional reference materials in their
tutorial manuals, on the web, and/or in related textbooks. Also, these
tools are all are a matter of convenience -- a general purpose
programming tool with some good libraries (e.g. Python + SciPy)
http://www.scipy.org/
http://www.scipy.org/Topical_Software
can let you do pretty much anything these specialized packages do and in
a much more expandable way -- it just takes *much* more time and
understanding to work with the general tools (though you learn a lot
doing that). I'd suggest perhaps doing both -- play with the tools *and*
try to learn how to program with various packages. :-)
As you can see from especially the third link above, there are several
other tools. I'm not sure how well all of these run under other
operating systems (most probably do run under Windows, but my wife and I
don't use that ourselves anymore except under emulation on the Mac or
Linux). You can probably find most of theses tools included (or easily
loadable) in Ubuntu (a version of GNU/Linux), which is available for
free here (you can download and burn a bootable CD to run these when you
want by rebooting, even if you run Windows):
http://www.ubuntu.com/
and they also mail out free CDs (but it may take some time -- see the
links on that page). There is also an "Edubuntu" educational variant,
but it seems more targeted at schools and I haven't tried it.
Instructions for loading some of these packages into Ubuntu:
http://ubuntuforums.org/showthread.php?s=c8e173f81d97693ff7bb02ad5cefba0a&t=402802&highlight=Mathematical+software
Any one of these packages is a project to get to know and understand. It
might takes quite a while to get a feel for them. But I think mastering
any of these packages will have a lot of practical value -- more so than
doing some paper exercises.
While our child is only three (and so I haven't tried to use these
packages to teach him math; I'm still focusing on John Holt's ideas of
seeing how small numbers like nine can be broken down and assembled many
ways), I have been involved tangentially with some people doing
mathematical education using computer software for more advanced math,
primarily using the programming language Python.
http://www.python.org/
There is a newer way of thinking about math and physics which suggests
thinking about that as just a subpart of computer programming, rather
than a more classical school subject approach which often puts computer
programming as an add-on under math & physics. :-) Essentially, by
developing programs and simulations of interest to a child, a child
learns more and more about basic principles of mathematics they need to
solve problems of interest to them, as well as to develop a certain type
of mathematical thinking skills related to precision and causality and
symbol manipulation and so on. Learning to program then becomes the
stepping stone to learning about math and physics.
For example, a child can learn a lot about trigonometry by just playing
around with a 2D or 3D turtle:
"turtle -- Turtle graphics for Tk using Python"
http://docs.python.org/lib/module-turtle.html
However they typically need to learn a little programming first (like
how to write a loop). Consider, for example, this college assignment:
"Mandelbrot, Turtle Graphics and L-Systems in Python"
http://www.cs.unc.edu/Courses/comp144-s02/assignments/2/index.html
I would think those sorts of experiments could keep a high schooler busy
for quite a while, while also learning about angles and distances and
mathematical reasoning (while they debug their graphical programs); but
I wouldn't try those as a first step for someone who did not already
know programming. Here is a simple code snippet for someone wanting to
learn some simple programming and turtle graphics with Python:
"Turtle Graphics (Python)"
http://www.daniweb.com/code/snippet304.html
Modify that to draw a few geometric shapes (cubes, octagons,
parallelograms, etc.) and it's a start down the path of 2D geometry. :-)
Here is a bigger set of links (mostly programs) related to
Python+Geometry for homeschoolers
http://www.4dsolutions.net/ocn/pygeom.html
In general the author of that site has a lot to say about the
intersection of computing and mathematics.
To get more advanced in 3D geometry, here is a free package by the late
Arthur Siegel to do interactive 3D geometry in Python:
http://pygeo.sourceforge.net/
"PyGeo is most fundamentally a framework for the creation of dynamic
geometric constructions - i.e. constructions which embody defined and
persistent geometric relationships responsive to real time on-screen
interactivity. PyGeo is, further, an implementation of this underlying
abstract framework - exposing a range of geometric objects as the
building blocks for virtual, dynamic geometric constructions. The focus
is away from Euclidian geometry and metrics, and toward later geometric
and mathematical developments - particularly those connected with
projective geometry of real space, and the geometry of complex numbers
on the plane and on the unit (Riemann) sphere."
Nothing about school-style trigonometry "proofs" there though -- except
the ones you build for yourself in passing to debug things. Here's lots
of links related to logic software and logic education if you want to
focus on proofs (includes some links to theorem provers):
http://www.cs.otago.ac.nz/staffpriv/hans/logiccourseware.html
But I would think this a dustier and drier route for most kids to get an
intuition for geometrical things than indirectly via programming.
Some fraction of the population has trouble visualizing or
rotating things in 2D or 3D,
http://en.wikipedia.org/wiki/Spatial_Visualization_Ability
so some kids find geometry much easier than others. I think 3D spatial
visualization tools may help narrow that gap, especially tools which can
make the 3d object rotate (at least slightly). Related exercises:
"Spatial Encounters: Exercises in Spatial Awareness."
http://www.womentechstore.com/edpw031-spatial-encounters-introduction.html
http://eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED236183&ERICExtSearch_SearchType_0=eric_accno&accno=ED236183
Likewise some kids almost need to visualize things to feel they
understand them as opposed to accept purely symbolic representations;
there are various approaches to mathematics that are more highly visual
than others. Example:
http://www.ms.uky.edu/~lee/visual05/visual05.html
If there is one issue that may make or break in some advanced math
studies, it is an ability to visualize. If the kid in question isn't
good at that, it may help more to focus early on visualization skills
(including blueprint reading) than to move ahead to more advanced topics.
Here is another programming language system Squeak (Smalltalk) which has
various educational tools in it like turtles:
http://www.squeak.org/Features/
It has a steeper learning curve than Python though.
In my own life I've found the only major use for trigonometry was
writing 3D graphics software. :-) Here is a example of such software
(available for free) which my wife and I wrote:
http://www.kurtz-fernhout.com/PlantStudio/index.htm
"PlantStudio Botanical Illustration Software is a tool for creating 3D
plant models and 2D illustrations. PlantStudio simulates herbaceous
(non-woody) plants like wildflowers and cut flowers, vegetables, weeds,
grasses, and herbs using a parameter-driven simulation of plant growth
and structure. You can "grow" plants over their life cycles, producing
lifelike images at any age. You can design, animate and breed a wide
variety of plants. By using the "evolutionary arts" of variation and
selection in the plant breeder, you can quickly and easily create whole
families of unique plants for your 3D scenes."
There's a lot of stuff in there about angles and rotations and so on --
can't help but learn a little about trigonometry (and botany) even by
just playing with it.
While I know schools tend to focus on a chain of algebra, trigonometry,
calculus, (especially for kids interested in engineering) I'd suggest
learning about probability and statistics (done as part of developing a
computer simulation) would generally be a more useful mathematical skill
in the real world for most people, since we all need to make decisions
every day related to risks and also trying to read between the lines of
newspaper articles and advertisements throwing statistics around in an
attempt to inform or persuade us.
For example, here is a fun link about how to lie with statistics:
http://faculty.washington.edu/chudler/stat3.html
"Ok...this is what you have been waiting for. How can you lie with
statistics? Actually, the purpose of this page is NOT to teach you how
to lie and cheat with statistics. Rather, I hope you will learn how it
is possible to be misled and how to spot "statistical abuse." You can
find poor use of statistics everywhere...magazines, newspapers, polls,
TV, even research papers. I do not want to hear of any of you readers
using these poor methods."
And here is a book on that topic:
_How to Lie With Statistics_
http://www.amazon.com/How-Lie-Statistics-Darrell-Huff/dp/0393310728
There are also a variety of free instructional videos available over the
web as part of the Annenberg CPB project if you have a fast internet
connection (and otherwise likely available from a local library or
through interlibrary loan):
http://www.learner.org/resources/browse.html
Exmaples:
"Against All Odds: Inside Statistics; This video instructional series
for college and high school classrooms and adult learners leads to a
greater understanding of statistics by exploring authentic examples —
from environmental studies to weight-loss programs."
"Algebra: In Simplest Terms; This video instructional series for college
and high school classrooms and adult learners guides students
step-by-step through algebra concepts, while highlighting common trouble
spots."
"Learning Math: Geometry; Learn the basics of geometry in this video-
and Web-based course for K-8 teachers."
I haven't watched any of those, but many years ago I did go through the
entire Mechanical Universe physics series for myself and liked it:
"The Mechanical Universe…and Beyond; This video instructional series
for college and high school classrooms and adult learners demystifies
physics and illustrates abstract concepts."
(Physics ends up being mostly math at more advanced stages.)
I also enjoyed their World of Chemistry, but I didn't see that is
online. There is also an accompanying textbook to the Mechanical
Universe; if anyone wanted something like a curriculum, that might
be a way to go which integrates mathematics into a physical setting that
helps motivate its study -- but you can also just watch the videos
for fun.
Here is another general resource for math education:
http://mathforum.org/
"The Math Forum is the leading online resource for improving math
learning, teaching, and communication since 1992.
_We are teachers, mathematicians, researchers, students, and parents
using the power of the Web to learn math and improve math education.
_We offer a wealth of problems and puzzles; online mentoring; research;
team problem solving; collaborations; and professional development.
Students have fun and learn a lot. Educators share ideas and acquire new
skills."
If your kid is a general theoretical math whiz
there is quite a bit of money to be won solving these incredible
difficult puzzles:
http://www.claymath.org/millennium/
"In order to celebrate mathematics in the new millennium, The Clay
Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven
Prize Problems. The Scientific Advisory Board of CMI selected these
problems, focusing on important classic questions that have resisted
solution over the years. The Board of Directors of CMI designated a $7
million prize fund for the solution to these problems, with $1 million
allocated to each."
It may even be fun just to look at the puzzles and think about what you
would need to know to even think about solving them. :-)
Here is a step down :-) -- the local William Lowell Putnam competition:
http://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition
http://math.scu.edu/putnam/
"Each year the annual Putnam Competition is held on the first Saturday
in December. ... The competition is open only to regularly enrolled
undergraduates, in colleges and universities of the United States and
Canada, who have not yet received a college degree. No individual may
participate in the competition more than four times. An eligible entrant
who is also a high school student must be informed of this four time limit."
Not much money even if you win the competition, but
perhaps a bit of glory. And something motivating for some -- to do well
in that contest. (Well, it helps that math departments sometimes pay for
a free lunch for the contestants. :-) Although ultimately the best
motivation comes from within for the love of the subject for itself.
Perhaps a competition might even be off-putting for some.
Anyway, even if you don't want to do the competition, the support
materials and sample questions may be of interest.
I'm a John Holt fan. Here is one thing he says about math:
http://www.naturalchild.org/guest/marlene_bumgarner.html
"What your philosophy about math?
...
The best way to meet numbers is in real life, as everything else. It’s
embedded in the context of reality, and what schooling does is to try to
take everything out of the context of reality. So everything appears
like some little thing floating around in space, and it’s a terrible
mistake. You know, there are numbers in building; there are numbers in
construction; there are numbers in business; there are numbers in
photography; there are numbers in music; there are fractions in cooking.
So wherever numbers are in real life, then let’s go and meet them and
work with them.
What subject matter do you see as essential?
None."
There is too much mathophobia in the world as it is -- in large part as
a consequence of how math is taught in schools:
http://www.google.com/search?hl=en&q=mathophobia
Yet higher math can be a very fun endeavor if done for its own sake.
As much as I like mathematical ideas, there are a lot of very successful
people in life who cannot do math at all. If your kid has other
interests, then advanced math is about as important as advanced zoology
-- probably less so. :-)
Anyway, plenty of open-ended free resources to play with and learn
something mathematical for kids or adults.
--Paul Fernhout
(Just joined the group.)
Fetteroll wrote:
> On Jun 14, 2007, at 11:20 AM, mommaofaton wrote:
>
>> I'm searching for suggestions, tips, or ideas for unschooling math.
>>
>
> It helps loads to not see math as something alien that needs learned
> but as a tool that is used while living life.
MOLLY
hello.
i'm molly. i thought i would introduce myself before i chime in on this topic.
i am unschooling my daughter and i agree, units or subject study can be a bore. as far as
your child comprehending the concept of numbers. i suggest cooking together. isadora
loves using her own measuring cups (we found the best heart shaped cups and spoons at
target after valentine's day clearance)
it is only natural that issy is curious about the numbers on the measuring spoons and
cups. we also do alot of gardening which develops a sense of counting and numbers.
i pretty much follow isadora's lead. we live on a farm so we tend to have many
opportunities at natural learning through daily living.
molly.karen
i'm molly. i thought i would introduce myself before i chime in on this topic.
i am unschooling my daughter and i agree, units or subject study can be a bore. as far as
your child comprehending the concept of numbers. i suggest cooking together. isadora
loves using her own measuring cups (we found the best heart shaped cups and spoons at
target after valentine's day clearance)
it is only natural that issy is curious about the numbers on the measuring spoons and
cups. we also do alot of gardening which develops a sense of counting and numbers.
i pretty much follow isadora's lead. we live on a farm so we tend to have many
opportunities at natural learning through daily living.
molly.karen
--- In [email protected], "Ren Allen" <starsuncloud@...> wrote:
>
> ~~You don't need to unschool math any more than you need to unschool
> the English language.~~
>
> Exactly.
> Seeing the world in subjects is an impediment to joyful unschooling.
> When math and history and science and everything else schools try to
> divide, are simply part and parcel of pursuing passions (yeah, I'm
> having fun with that one today), then subjects become irrelevant. The
> interest and fascination humans have with different topics and
> information will involve math, as we follow those interests.
>
> We don't need to worry about getting math into our children. We need
> to focus on living a rich and full life.
>
> It IS good to be aware if we ourselves have hangups and damage about
> math (or anything else) so we don't automatically shun things that are
> related to it. I found that as I healed old school wounds, I saw math
> in a totally different light. Math is everywhere. It's in art, music
> and daily life. It can't be avoided.
>
> Ren
> learninginfreedom.com
>
Joanne
Hi Kay...
Unschooling Math was a topic a few months back on Unschooling
Voices. I'm sure you'll find some ideas and also some ways to change
your own perspective on math. Here's the link:
http://tinyurl.com/26pt6x
Scroll down the list until you see it. I think it was September or
October.
~ Joanne ~
Unschooling:
http://anunschoolinglife.blogspot.com
Adoption:
http://foreverparents.blogspot.com
--- In [email protected], "mommaofaton"
<mommaofaton@...> wrote:
Unschooling Math was a topic a few months back on Unschooling
Voices. I'm sure you'll find some ideas and also some ways to change
your own perspective on math. Here's the link:
http://tinyurl.com/26pt6x
Scroll down the list until you see it. I think it was September or
October.
~ Joanne ~
Unschooling:
http://anunschoolinglife.blogspot.com
Adoption:
http://foreverparents.blogspot.com
--- In [email protected], "mommaofaton"
<mommaofaton@...> wrote:
>unschooling
>
> Hello, I'm searching for suggestions, tips, or ideas for
> math. Any suggestions would be appreciated. Thank you.
>
> Kay Beth
> Country Momma
>
Meredith
--- In [email protected], "Paul D. Fernhout"
<pdfernhout@...> wrote:
programming is s/he is *doing* real mathmatical thinking for its own
sake - not as a "stepping stone". Sure its possible to build on that
knowledge, but that's true of any kind of real knowledge.
My stepson is learning about programming while playing Runescape.
He's recently learned how to create a "bot" to look up other
players' stats before he picks a fight with them. He's not
interested in "stepping stones" or thinking about what he's doing
as "mathmatical". He's choosing to learn to program bc it furthers
another goal entirely.
*your kid* focusing on? Was it his idea to break numbers down and
reassemble them? Now is a great time to step back and see what he's
learning and how without imposing expectations about how math
*should* look or be learned. You may be surprised to discover the
bredth and sophistication of his abilities.
I love reading theories about learning in general and mathematical
learning in particular, but I've also found it helpful to look at
that information as a way of understanding what my child may be
doing, rather than trying to "get" her to learn in a certain manner.
anything in the process of learning and understanding. Some people
visualize words as they are spoken. Others form visual images
relating to the words, but not the actual words. That the sort of
thing that fascinates me ;)
it were. Why would someone who wasn't terribly visual be drawn to
this kind of advanced math in the first place? That could very well
play in to the "make or break" aspect.
go about learning things in general so we can help them find the
tools that work best for them. But again, if someone didn't have a
strong ability to visualize, and wasn't being pushed or "encouraged"
to study higher math, I find myself wondering why he or she would
want to do that? Maybe to further some other goal (like my stepson
with the programming) but that assumes there's only one route to a
given goal. That's not the kind of thinking unschoolers tend to
develop.
---Meredith (Mo 5.5, Ray 13)
<pdfernhout@...> wrote:
>> Essentially, byneed to
> developing programs and simulations of interest to a child, a child
> learns more and more about basic principles of mathematics they
> solve problems of interest to them, as well as to develop acertain type
> of mathematical thinking skills related to precision and causalityand
> symbol manipulation and so on. Learning to program then becomes theOr not. One of the valuble aspects of a kid choosing to learn
> stepping stone to learning about math and physics.
programming is s/he is *doing* real mathmatical thinking for its own
sake - not as a "stepping stone". Sure its possible to build on that
knowledge, but that's true of any kind of real knowledge.
My stepson is learning about programming while playing Runescape.
He's recently learned how to create a "bot" to look up other
players' stats before he picks a fight with them. He's not
interested in "stepping stones" or thinking about what he's doing
as "mathmatical". He's choosing to learn to program bc it furthers
another goal entirely.
> While our child is only three (and so I haven't tried to use theseideas of
> packages to teach him math; I'm still focusing on John Holt's
> seeing how small numbers like nine can be broken down andassembled many
> ways)This may sound snarky, but its intended as a real question - what's
*your kid* focusing on? Was it his idea to break numbers down and
reassemble them? Now is a great time to step back and see what he's
learning and how without imposing expectations about how math
*should* look or be learned. You may be surprised to discover the
bredth and sophistication of his abilities.
I love reading theories about learning in general and mathematical
learning in particular, but I've also found it helpful to look at
that information as a way of understanding what my child may be
doing, rather than trying to "get" her to learn in a certain manner.
> Likewise some kids almost need to visualize things to feel theySome people are certainly more visual and will visualize most
> understand them as opposed to accept purely symbolic
>representations;
anything in the process of learning and understanding. Some people
visualize words as they are spoken. Others form visual images
relating to the words, but not the actual words. That the sort of
thing that fascinates me ;)
> If there is one issue that may make or break in some advanced mathIts interesting to conceptualize this from the other direction, as
> studies, it is an ability to visualize.
it were. Why would someone who wasn't terribly visual be drawn to
this kind of advanced math in the first place? That could very well
play in to the "make or break" aspect.
>If the kid in question isn'tFrom an unschooling perspective, its helpful to notice how our kids
> good at that, it may help more to focus early on visualization
>skills
go about learning things in general so we can help them find the
tools that work best for them. But again, if someone didn't have a
strong ability to visualize, and wasn't being pushed or "encouraged"
to study higher math, I find myself wondering why he or she would
want to do that? Maybe to further some other goal (like my stepson
with the programming) but that assumes there's only one route to a
given goal. That's not the kind of thinking unschoolers tend to
develop.
> Anyway, plenty of open-ended free resources to play with and learnAlways glad for more free stuff, thanks!
> something mathematical for kids or adults.
---Meredith (Mo 5.5, Ray 13)
Paul D. Fernhout
Meredith wrote:
And I can agree you are right -- look at my implicit thinking in even
using the phrase "to teach him math" instead of, say, "to help support
his desire to learn math".
Right now, at three, he is focused on issues like moving quantities of
sand and water around in various ways. And what you say makes sense; I
should respect that -- even if he is not too sure consistently on what
number word comes after three and has not much patience or interest in
games involving counting (like Chutes and Ladders). :-)
Consider for example:
"What can children learn from water play?"
http://www.communityplaythings.com/c/Resources/Articles/SandandWater/MakingtheMostofWaterPlay.htm
"Without the time and opportunity for lots of exploration, a child
formulates fewer meaningful concepts. While water play promotes
problem-solving and thinking skills in general, it is particularly well
suited to the development of concepts in mathematics and science."
Some concepts they list there learned in water play include:
"empty/full many/ few before/ after
thick/ thin more/ less same/ different
heavy/ light shallow/ deep greater/ less than
sets classification rational counting
liquid measure ordinal counting linear measure".
Another article on that topic:
"Playing 'better than lessons'"
http://news.bbc.co.uk/1/hi/education/4456131.stm
"A study found that five-year-olds found it difficult to adapt to the
formal curriculum. The research, from the National Foundation for
Educational Research, concluded that the time spent sitting still should
be reduced. It said that children should have more access to
"play-based" learning. The study suggested that schools should "allocate
resources to enable children to experience some play-based activities
that give access to opportunities such as sand and water, role play,
construction and outdoor learning"."
Another related article:
http://www.timesonline.co.uk/tol/news/article721863.ece
"After studying 25,000 children across both state and private schools
Philip Adey, a professor of education at King’s College London
confidently declares: “The intelligence of 11-year-olds has fallen by
three years’ worth in the past two decades.” ... So why are children now
doing so badly? Possible explanations are numerous. Youngsters don’t get
outside for hands-on play in mud, sand and water — and sandpits and
water tables have been squeezed out in many primary schools by a
relentless drilling of the three Rs and cramming 11- year-olds for the
national tests. “By stressing the basics — reading and writing — and
testing like crazy you reduce the level of cognitive stimulation.
Children have the facts but they are not thinking very well,” says Adey.
“And they are not getting hands-on physical experience of the way
materials behave.” "
Thanks for the reminder; hard to break old habits of schooling, even
when you are trying. :-(
I do feel, especially in math, there is the issue of a kid not knowing
what he or she does not know -- that is why we often feel less smart as
we get older :-) as we learn more about our own ignorance, and why our
parents often suddenly seem much more intelligent after we have kids of
our own. :-) But that issue of "unknown unknowns" is more applicable to
older ages (perhaps teens) where dealing with abstraction becomes
important in pursuing advanced math.
Thanks again for insightful your reply.
--Paul Fernhout
> --- In [email protected], "Paul D. Fernhout"Thanks for sharing your paradigm shifting thoughts.
> <pdfernhout@...> wrote:
>> While our child is only three (and so I haven't tried to use these
>> packages to teach him math; I'm still focusing on John Holt's
> ideas of
>> seeing how small numbers like nine can be broken down and
> assembled many
>> ways)
>
> This may sound snarky, but its intended as a real question - what's
> *your kid* focusing on? Was it his idea to break numbers down and
> reassemble them? Now is a great time to step back and see what he's
> learning and how without imposing expectations about how math
> *should* look or be learned. You may be surprised to discover the
> bredth and sophistication of his abilities.
And I can agree you are right -- look at my implicit thinking in even
using the phrase "to teach him math" instead of, say, "to help support
his desire to learn math".
Right now, at three, he is focused on issues like moving quantities of
sand and water around in various ways. And what you say makes sense; I
should respect that -- even if he is not too sure consistently on what
number word comes after three and has not much patience or interest in
games involving counting (like Chutes and Ladders). :-)
Consider for example:
"What can children learn from water play?"
http://www.communityplaythings.com/c/Resources/Articles/SandandWater/MakingtheMostofWaterPlay.htm
"Without the time and opportunity for lots of exploration, a child
formulates fewer meaningful concepts. While water play promotes
problem-solving and thinking skills in general, it is particularly well
suited to the development of concepts in mathematics and science."
Some concepts they list there learned in water play include:
"empty/full many/ few before/ after
thick/ thin more/ less same/ different
heavy/ light shallow/ deep greater/ less than
sets classification rational counting
liquid measure ordinal counting linear measure".
Another article on that topic:
"Playing 'better than lessons'"
http://news.bbc.co.uk/1/hi/education/4456131.stm
"A study found that five-year-olds found it difficult to adapt to the
formal curriculum. The research, from the National Foundation for
Educational Research, concluded that the time spent sitting still should
be reduced. It said that children should have more access to
"play-based" learning. The study suggested that schools should "allocate
resources to enable children to experience some play-based activities
that give access to opportunities such as sand and water, role play,
construction and outdoor learning"."
Another related article:
http://www.timesonline.co.uk/tol/news/article721863.ece
"After studying 25,000 children across both state and private schools
Philip Adey, a professor of education at King’s College London
confidently declares: “The intelligence of 11-year-olds has fallen by
three years’ worth in the past two decades.” ... So why are children now
doing so badly? Possible explanations are numerous. Youngsters don’t get
outside for hands-on play in mud, sand and water — and sandpits and
water tables have been squeezed out in many primary schools by a
relentless drilling of the three Rs and cramming 11- year-olds for the
national tests. “By stressing the basics — reading and writing — and
testing like crazy you reduce the level of cognitive stimulation.
Children have the facts but they are not thinking very well,” says Adey.
“And they are not getting hands-on physical experience of the way
materials behave.” "
Thanks for the reminder; hard to break old habits of schooling, even
when you are trying. :-(
I do feel, especially in math, there is the issue of a kid not knowing
what he or she does not know -- that is why we often feel less smart as
we get older :-) as we learn more about our own ignorance, and why our
parents often suddenly seem much more intelligent after we have kids of
our own. :-) But that issue of "unknown unknowns" is more applicable to
older ages (perhaps teens) where dealing with abstraction becomes
important in pursuing advanced math.
> I love reading theories about learning in general and mathematicalSounds like a great perspective.'
> learning in particular, but I've also found it helpful to look at
> that information as a way of understanding what my child may be
> doing, rather than trying to "get" her to learn in a certain manner.
Thanks again for insightful your reply.
--Paul Fernhout
Meredith
--- In [email protected], "Paul D. Fernhout"
<pdfernhout@...> wrote:
workshops...and now have a teen stepson and I've often found just
the opposite. Most people leave school with such a narrow idea of
what "math" is and *that* hampers their understanding. When I've
been able to find ways to explain concepts that don't look or sound
like "math" I find many people have a deeper understanding than they
themselves realize - its not that they don't know what they don't
know, they don't know or value what they *do* know.
I think the "less smart" feeling that some people have is a result
of having developed anxiety about learning. In quilting workshops
I've seen (and experienced myself) the delight that comes over
people when they are learning something entirely new. The difference
is that these are people who are choosing to do something for the
enjoyment of it and learning as a result. That's a basic principle
of unschooling - that natural learning is enjoyable.
Right now I'm learning to juggle (just got some clubs, woooo hooo!).
There are definately times when I feel like I'm all thumbs, or that
I'm making infintessimal progress. But I don't feel "less smart" for
not knowing how to juggle. I *do* get a sense of anxiety around
certain jugglers, who have a very didactic way of talking to me
about juggling. Other jugglers I know are simply supportive -
they'll offer help if I seem like I need it, applaud my sucesses and
invite me to laugh at my goofs. I don't have any anxiety at all
about juggling in front of them.
explore the idea. I think the folks I'm describing as "didactic" are
making a point of telling me how ignorant I am. Not in a mean way. I
can tell these guys - its just two - are trying to be helpful, even
encouraging. But somehow it leads to me feeling worse about my
ability.
The jugglers I know who are being supportive, but not didactic,
aren't any less skilled jugglers than these two guys - if anything,
they're better. Watching them, I can *see* all the things I can't do
yet, but I'm left with a sense of "hey, wow, can I do that,
someday?" instead of "I'm a long way from there".
other things - one of the reasons I brought up juggling, although
its pretty easy to argue that juggling is math ;) Learning just
isn't segmented into clear categories. Individual tendencies,
development, past experiences, and the environment all interact and
overlap and swirl around. Knowing what I know about my own patterns
of learning is far more useful to *me* in getting over the first
stage of learning *anything* than any other skill or piece of
information.
---Meredith (Mo 5.5, Ray 13)
<pdfernhout@...> wrote:
>> I do feel, especially in math, there is the issue of a kid notknowing
> what he or she does not know -- that is why we often feel lesssmart as
> we get olderOver the years I've tutored math, and also taught quilting
workshops...and now have a teen stepson and I've often found just
the opposite. Most people leave school with such a narrow idea of
what "math" is and *that* hampers their understanding. When I've
been able to find ways to explain concepts that don't look or sound
like "math" I find many people have a deeper understanding than they
themselves realize - its not that they don't know what they don't
know, they don't know or value what they *do* know.
I think the "less smart" feeling that some people have is a result
of having developed anxiety about learning. In quilting workshops
I've seen (and experienced myself) the delight that comes over
people when they are learning something entirely new. The difference
is that these are people who are choosing to do something for the
enjoyment of it and learning as a result. That's a basic principle
of unschooling - that natural learning is enjoyable.
Right now I'm learning to juggle (just got some clubs, woooo hooo!).
There are definately times when I feel like I'm all thumbs, or that
I'm making infintessimal progress. But I don't feel "less smart" for
not knowing how to juggle. I *do* get a sense of anxiety around
certain jugglers, who have a very didactic way of talking to me
about juggling. Other jugglers I know are simply supportive -
they'll offer help if I seem like I need it, applaud my sucesses and
invite me to laugh at my goofs. I don't have any anxiety at all
about juggling in front of them.
> that is why we often feel less smart asI'm really glad you posted this, since its giving me a chance to
> we get older :-) as we learn more about our own ignorance
explore the idea. I think the folks I'm describing as "didactic" are
making a point of telling me how ignorant I am. Not in a mean way. I
can tell these guys - its just two - are trying to be helpful, even
encouraging. But somehow it leads to me feeling worse about my
ability.
The jugglers I know who are being supportive, but not didactic,
aren't any less skilled jugglers than these two guys - if anything,
they're better. Watching them, I can *see* all the things I can't do
yet, but I'm left with a sense of "hey, wow, can I do that,
someday?" instead of "I'm a long way from there".
>especially in mathI don't find learning "math" necessarily different from learning
other things - one of the reasons I brought up juggling, although
its pretty easy to argue that juggling is math ;) Learning just
isn't segmented into clear categories. Individual tendencies,
development, past experiences, and the environment all interact and
overlap and swirl around. Knowing what I know about my own patterns
of learning is far more useful to *me* in getting over the first
stage of learning *anything* than any other skill or piece of
information.
---Meredith (Mo 5.5, Ray 13)